Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Ronson and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of Probability and theoretical probabilities.

This lesson is called Problem Solving with Theoretical Probabilities, and by the end of today's lesson, we'll be able to apply our knowledge of probability to help us solve problems. Here are some previous keywords that we're gonna use again during today's lesson.

So you may wanna pause the video at this point if you want to remind yourselves what any of these words mean, and then press play when you're ready to continue.

This lesson contains two learn cycles.

In the first learn cycle, we're gonna be considering one of the main uses of probability, and that is using it to inform our decisions.

Then the second learn cycle, we'll be thinking about experiments and how many trials may be required in order to observe a difference in the number of outcomes.

Let's start off though with using probability to inform decisions.

Here we have Andeep and Laura.

They are playing a game that involves choosing two dice, rolling them, and summing the numbers.

Now there are four dice they can choose from.

Two of the dice are cubes containing the numbers 1 to 6, and the other two dice are octahedrons containing the numbers 1 to 8.

They have to choose two out of these four dice to roll.

Andeep says, "I want my two dice to sum to 8." So Laura tries to help him.

Laura says, "There will be more ways to roll a total of 8 if you use the octahedron dice, but there'll also be more ways to roll other numbers too." So this decision is not quite as straightforward as it might seem.

Laura says, "I wonder which representation of the outcomes will be best suited to model this problem?" What do you think about that? And which combination of two dice should and Andeep choose? Pause the video while you think about this, and then press play when you're ready to talk about it together.

Let's think about this together now.

Some representations that we've used before to represent outcomes include outcome trees, two-way outcome tables, and Venn diagrams. If we try to use a tree for this situation, we would need to have six branches each time we use the cube dice and eight branches each time we use octahedron dice.

So that's not gonna be so straightforward to use.

And the Venn diagram would be helpful if we were thinking about grouping outcomes together into categories such as multiples or factors and so on, but that's not really what we're doing here.

The most suitable one would be to use a two-way outcome table.

So the outcomes at each pair of dice can be represented in a two-way outcome table that looks a bit like this.

The top row shows us the numbers 1 to 8 for one of the dice and the left hand column, show the numbers 1 to 8 for the other dice.

And then if we roll these two dice and sum the numbers we get.

here are all the different possible outcomes we could get.

So if we want to use this table to find the probability that the dice total 8, we could look at the number of ways that could happen.

There are seven ways of getting 8, and the number of outcomes altogether.

There are 64 outcomes altogether, so the probability would be 7 over 64.

If we had the 8-sided dice and the 6-sided dice, our table would look a bit more like this.

Then the probability of getting an 8 would be 6 over 48, 640 eights.

And if we used two 6-sided dice, the table would look like this, and the probability of getting 8 would be 5, 36.

So now we have our three probabilities.

Our next job is to compare them and see which probability is the greatest because that would tell us which pair of dice would give us the best chance of getting what we want.

Now to compare these probabilities, we could find a common denominator or a common numerator, but that could lead to quite large numbers and some tricky calculations.

Andeep say, "It would be easier to compare the probabilities if I convert them all into decimals", and so he does.

Now we can see these three probabilities as decimals.

We can compare them and see that 0.

138 recurring is the greatest so using two cube dice would give us the best probability of rolling a total of 8 in this situation.

Now after watching Andeep do this, Laura says, "You didn't have to draw all three separate tables to solve this.

It could have been solved by just drawing a single table." Can you think what Laura might mean? Pause the video while you think about this and press play when you're ready to continue.

Laura says, "You could start by drawing the biggest table that was required and then ignore the parts you don't need later on." So here we have a table for using two 8-sided dice and we can get the probability.

in that case would be 7/64.

But then when we wanna find the probability of using a 8-sided dice and a 6-sided dice, we could just ignore those last two columns and then find the probability from there, which we can see is 6 over 48 or 6/48.

And then when we want to think about just using two 6-sided dice, can you see which part of the table we would ignore then? We can ignore the bottom two rows, and then find the probability of getting 8 with these two dice is 5/36.

So let's check what we've learned.

We have two spinners, each numbered 1 to 3.

Each spinner is spun and the outcomes are added together.

The two-way table shows the sample space.

So what's the probability of getting a total of 4? Pause the video while you work this out and press play when you're ready to continue.

The probability is 3/9.

We can see that because there are 3 outcomes which have a total of 4, and there are 9 outcomes altogether.

Now we have a two-way outcome table, and you'll notice that some of it is highlighted.

Which combination of spinners is represented by the highlighted part of the sample space? Your options are A, B, and C.

Pause the video while you choose your answer and then press play when you're ready to see the answer.

The answer is B.

We have a 1 to 3 spinner, and that is represented by the columns 1, 2, and 3.

And we have a 1 to 5 spinner, and that is represented by the rows 1, 2, 3, 4, and 5.

So now which combination of spinners gives the best probability of getting a total of 4? You'll need to do a little bit of working out for this question, so pause the video while you work through it and press play when you're ready for an answer.

Well, here are the probabilities of getting a 4 with each pair of spinners, and now we've got these probabilities, we need to compare them.

And what's quite helpful here is that the numerator is the same in each fraction, so we don't necessarily need to change anything, we can just compare the denominators.

and by doing that we can see that 3/9 is the greatest fraction, so that means it's the greatest probability.

Okay, it's over to you now for task A.

This task contains one question and here it is.

"Andeep is playing a game that involves choosing two dice, rolling them and summing the numbers.

Two of the dice are cubes, container numbers 1 to 6 and the other two dice are tetrahedron, that's four-sided shape, containing the numbers 1 to 4." You need to decide how to model this situation using some kind of representation of the outcomes, and then use that representation to help you answer question A, B, C, and D.

With parts A, B, and C, you've got to work out which combination of dice gives you the greatest probability of rolling each of those totals.

Then part D says, "Write a summary that advises a player how to decide which pair of dice gives them the greatest probability for the total they want." So you've got part A, B, and C to help you with that, but those are only three possible outcomes when you roll these two dice together and sum them.

What about all the other possible outcomes? How does a player decide which pair of dice is best in which situation? So for Part D, you may wanna write a few sentences that describe what to do in different situations.

When should you use each combination of dice? Pause the video while you have a go at this and press play when you're ready to work through some answers.

Okay, let's see how we got on with that.

So we could represent this situation using a two-way outcome table, and it would look a bit like this.

And we now use this table to help us answer some questions.

"Which combination of dice gives the greatest probability of rolling a total of 8?" Well, here is the probability of that with each combination of dice, and we can see that the greatest probability is when we use 2 cube dice, two 6-sided dice.

"Which combination of dice gives the greatest probability of rolling a total of 6?" Well, here's the probability with each combination of dice, and this time we can see the greatest probability is when using 2 tetrahedron dice, two 4-sided dice.

"Which combination of dice gives the greatest probability of rolling a total of 7?" Well, here are the probabilities.

In this case, you can see that the greatest probability appears twice, either if you use 2 cube dice or if you use 1 of each dice.

And then we need to write a summary that advises a player how to decide which pair of dice gives 'em the greatest probability of rolling what they want.

Let's think about this.

"We could use two tetrahedron dice when you want to roll a total of 6 or less.

You could use the two cube dice when you want to roll a total of 7 or more.

Alternatively, you could use one of each dice if you want to roll a total of 7." That's the only situation where one of each dice is most helpful.

Well done so far.

Now let's look at the number of trials required to observe a difference.

"Here we have Lucas and Sofia who have designed an app containing a chance-based game.

When players are playing this game, they have a 1% probability of finding a rare gem each turn.

This means a player could expect to find approximately 1 gem in every 100 turns.

Now Lucas changes the game to make it a bit easier for players to find a gem." Let's hear it from Lucas.

He says, "I've increased the probability of finding a gem by 50%." Sofia says, "Oh no! That makes it far too easy.

Players will now have approximately 51 gems for every 100 turns." Is Sofia correct? Pause the video while you think about what's going on here and what Sofia has said, and then press play when you're ready to continue.

Well, let's work through this step-by-step.

"The probability of finding a gem in the original version was 1%.

Lucas increased the probability by 50%." Now if we do that percentage increase, this is what calculation would do, and this is what we'd get.

The 1% here is the original probability.

The 1.

5 represents the 50% increase that was applied to that probability, and then the 1.

5% is the new probability.

So the probability of finding a gem in the new version of the game is 1.

5%.

Now, this situation could be a little bit confusing because we are increasing percentages by percentages, but Lucas says, "This calculation could also be performed with the probabilities expressed as their equivalent decimals." Let's look at that together.

The original probability was 0.

01, and then Lucas increased that probability by 50%.

Here's our calculation.

0.

01, which is the original probability.

multiplied by 1.

5, which is that 50% increase, equals 0.

015, which is the new probability.

That means the probability of finding a gem in the new version of the game is 0.

015.

And Sofia says, "The increase isn't as much as I thought." "They then consider what difference the increased probability will make to the number of gems that players can expect with each version of the game." So we have a table here where we can see the probability of finding a gem in the original version was 1%, in the new version was 1.

5%, and the difference between those two probabilities is 0.

5%.

"They compare the number of gems that could be expected from 100 turns." So that would be these values here.

In the original version, you would expect to get one gem every 100 turns.

In the new version, you expect to get 1.

5 gems in every 100 turns, and the difference is half a gem.

But that doesn't quite make sense.

Lucas says, "You can't find 1.

5 gems." So the difference between these two versions isn't even a whole gem yet, for when you have 100 turns.

So with that in mind, they compare the number of gems that can be expected from 1000 turns.

So in the original version, 1% of 1000 can be calculated by doing 0.

01 X 1000, which is 10.

In the new version, we can do 1.

5% of 1000 by doing 0.

015 X 1000, which is 15.

And the difference between that is 15, subtract 10, which is 5.

So if players took 1000 turns using each version of this game, they could expect five more with the new version than they could with the original version.

Sofia says, "When you said the probability had increased by 50%, it first sounded like players were gonna get loads more gems, but even after 1000 turns, players can only expect to find 5 extra gems with the new version of the game.

That's not many gems." So I wonder what the minimum number of turns a player would need to take to expect an actual difference between the number of gems getting in the new version compared to the original version.

They consider the number of turns required to expect a difference of 1 extra gem between the two versions of the game.

So we have the number 1 now in the bottom right hand box, which is the difference between the number of gems expected in the new version compared to the original one.

And what we wanna work out is what is the value of x, where X represents the number of turns needed.

Let's think about this together then.

With the original version, the probability is 1%.

So to figure out how many gems to expect, we'll do 1% of the number of turns, so 1% of X, we can write that as an expression 0.

01x, and for the new version, the probability is 1.

5%.

So if we will work out the number of gems expected, we do 1.

5% of the number of turns, which is X, so that is no 0.

015x.

And what we want to know is when is the difference between these two expressions equal to one? So we can make an equation that looks a bit like this, simplify it, and solve it to find the value of X, which is 200.

So that means we would only expect to observe a difference between the two versions of the game after a player has taken 200 turns.

You'll take 200 turns to expect a difference of one between the new version and the original version.

So let's check what we've learned there.

Here we have again an original game and a new game.

In the original game, the probability of finding a gem is 5% and it says the probability of finding a gem has now increased by 40% from the original game to the new game.

So what is the probability of finding a gem in the new game? Pause the video while you make a choice and then press play when you're ready for an answer, The answer is B, 7%, and we can get that by taking the original probability, which is 0.

05, which is equivalent to 5%, and then multiplying it by 1.

4, that's to 40% increase, that gives us 0.

07, which is 7%.

We've now expanded a table, so we have a column for the difference and also a row for the expectation per 100 turns.

The question says, "d is the difference between the number of gems that can be expected after 100 turns with each game.

What is the value of d?" Pause the video while you work this out and then press play when you're ready for an answer.

The answer is two.

We'd expect 5 gems for every 100 turns in the original game, we'd expect 7 gems for every 100 turns in the new game, and the difference between those is 2.

Or you can find the difference between the percentages, which is 2% and then do 2% of 100.

Let's expand the table even more now.

X is a number of turns required to expect a difference of 1 gem between the 2 games.

What is the value of x? Pause the video while you have a go at this and press play when you're ready for an answer.

Well, we could write an expression for how many we'd expect after 100 turns for each game, that would be 0.

05x for the original game, and 0.

07x for the new game.

We could then set up an equation where the difference between these two expressions is equal to 1, simplify the equation and then solve the equation to get x = 50.

Okay, it's over to you now for task B.

This task contains two questions, and here is question one.

"Lucas creates two versions of the same video game, version A and version B.

Version B is just a tweaked version of version A.

In version A, the probability of a player finding a gem is 0.

02.

Players are 10% more likely to find a gem in version B than they are in version A.

So you need to first complete this table and then work out how many turns would it take to expect a difference of one gem between the two versions of the game.

Pause the video while you work through this and then press play when you're ready for question two.

And here is question two.

"Sofia creates a video game where players have a 0.

5% probability of getting caught in a trap each turn.

You don't wanna get caught in a trap.

She creates a new version of the game where players are now 20% more likely to get caught in a trap each turn compared to the original version of the game.

So what is the probability of getting caught in a trap in the new version of the game? And then how many turns would it take to expect the difference of one extra trap between the two versions of the game?" You can draw a table for this one if you want to work through it in the same way as we have done previously, but you don't have to.

So pause the video while you work through it and press play when you're ready to work through some answers.

So let's see how we got.

In question 1, we have two versions of the same game, where in version A, the probability of finding a gem is 0.

02.

And in version B, you are 10% more likely to find a gem than in version A and we have to complete the table.

Well, the probability of finding a gem in version B is 0.

022, and the difference between these probabilities is 0.

002.

We expect a number of gems after 100 turns is 2 for version A, 2.

2 for version B, and as a difference there of 0.

2 gems. Now, 2.

2 gems and 0.

2 gems don't make practical sense, which is why it's helpful to look at creating a number of turns, like 1000 turn.

After 1000 turns, in version A, you can expect 20 gems. In version B, 22 gems and a difference of 2 gems between the two versions of the game.

"So how many turns would it take to expect a difference of one gem between the two versions of the game?" Well, we could let XB, the minimum number of turns required, and then write an expression for how many gems we'd expect with version A, which would be 0.

02x, how many gems we'd expect for version B, which would be 0.

022x, and then set up an equation with a difference between these two expressions is one, simplify the equation and solve it to get 500 turns.

An alternative way to have done this could be to look at how the difference for 1000 gems was two, and we wanna get a difference of one.

So we need to divide by two to get from that row to the row below, which means divide the number of turns by two, 1000/2 is 500.

And then in question two, with Sofia's video game, players initially have a 0.

5% probability of getting caught on a trap each turn.

This probability is then increased by 20%.

So what is a probability getting caught on a trap in the new version of the game? That's 0.

5% X 1.

2 for your 20% increase, so that gives you 0.

6%.

So the probability of getting caught on a trap was not 0.

5% and is now not 0.

6%.

So how many turns would it take to expect a difference of one extra trap between the two versions of the game? Well, here's our equation.

We can simplify it and solve it to get X equals 1000.

So between the original version of the game and the new version, whereas an increased probability of getting caught a trap, we can only expect a difference of 1 extra trap after 1000 turns with this version of the game.

Great work today.

Let's now summarise what we've learned in this lesson.

"Knowledge of probability can be applied to real world situations such as games of chance like we've seen in today's lesson, but other situations as well.

And in these situations, probability can be used to aid decision making." If there's a particular outcome we want to achieve, by considering the probabilities of achieving that outcome with different choices that we make, we can make the decision that gives us the best chance of achieving that outcome.

It doesn't guarantee it'll happen, but it gives us the greatest likelihood of it happening out of those decisions.

"Creating a sample space by using either a table or a diagram may help you solve a problem involving probability." And you may need to choose which type of representation is best suited for the problem you're trying to solve.

"Theoretical probabilities come from long-term behaviour and therefore they may not necessarily be seen in the short term, and a very large number of trials may be required in order to observe a difference between events with different probabilities." Just because two events have different probabilities doesn't necessarily mean that one of those will always happen more often than the other, or if a probability has been increased, You need to think about how many trials is required to actually see the difference between what it was, and what it is now after the increase.

Thank you very much.

Have a great day.